Mechanical Detection of the De Haas–van Alphen Effect in Graphene

In our work, we study the dynamics of a graphene Corbino disk supported by a gold mechanical resonator in the presence of a magnetic field. We demonstrate here that our graphene/gold mechanical structure exhibits a nontrivial resonance frequency dependence on the applied magnetic field, showing how this feature is indicative of the de Haas–van Alphen effect in the graphene Corbino disk. Relying on the mechanical resonances of the Au structure, our detection scheme is essentially independent of the material considered and can be applied for dHvA measurements on any conducting 2D material. In particular, the scheme is expected to be an important tool in studies of centrosymmetric transition metal dichalcogenide (TMD) crystals, shedding new light on hidden magnetization and interaction effects.


Thermodynamic analysis and forces acting on a movable capacitor (finite DOS)
Let us consider the energy per unit area of a capacitor with one of the plates constituted by a finite-DOS material (in our case graphene), in the presence of an external magnetic field B U (n, B) = 1 2 e 2 n 2 C g + Ξ (n, B) . (S1) The first term in Eq. (S1) is the energy associated with the electrical field building up between the plates of the capacitor, whereas the second corresponds to the energy related to the finite density of states (DOS) of the system. In this description, the charge n on the capacitor plates and the magnetic field B are the control parameters.
Considering a standard thermodynamic relation, 1 we can define the electrochemical potential asμ = ∂U ∂n z,B = e 2 n C g + µ (n) (S2) Figure S1: Top. Lumped-elements description of a capacitor with one plate constituted by a finite-DOS material (yellow) to which a voltage V g is applied. Bottom. The voltage drop leads to the buildup of an electric field between the capacitor plates, corresponding to an electrostatic potential drop ϕ. Owing to the finite DOS, part of the energy provided by the voltage source goes into promoting electrons to higher-energy single-particle states (increase of the chemical potential µ).
implying n = Cg e 2 (μ − µ). This relation, written as allows us to show that, for C g → ∞, the tunability of the electrochemical potentialμ is directly translated into the tunability of the chemical potential of the membrane µ. Hereμ is the electrochemical potential consisting of the electrostatic eϕ = e 2 n Cg and chemical µ = ∂Ξ ∂n potentials. 2 We can now consider the thermodynamic potential Ω (μ) whereμ = eV g is the independent control parameter determined by an external voltage source V g , along with B. From equation (S1), we obtain Ω (μ, B) = U (n, B) −μn = 1 2 where we have defined Φ (μ, B) . = Ξ (n (μ, B)). Using equation (S3), we can calculate the force for constantμ, i.e. at constant external applied voltage To this end, let us obtain a preliminary result where C ′ g = ∂C g /∂z. Equation (S5) implies, in addition, that and that the electrostatic force per unit area can be expressed as which is consistent with the expression given in the literature about carbon nanotubes. 3,4 As discussed in the main text, the distinguishing factor between the full thermodynamic potential Ω (μ = eV g , B) and that of the graphene disk Ω disk (x, B) is the control parameter µeV g versus µ or n. In  Figure S2: a) Magnetization oscillations obtained from equation S3 using v F = 10 6 m/s and τ q = 0.19 ps at fixed µ = 31 meV (red) and µ(B) for C g ≈ 1.15 × 10 −5 F/m 2 , V g = 9.8V (blue). The latter case, for these parameters, essentially corresponds to a charge-controlled situation, with n ≈ 7 × 10 −14 m −2 . Lorentzian broadening with the same disorder has been considered for both plots. b) The oscillating part χ m,osc of magnetic susceptibility χ m (µ, B) for the same parameters as in panel a).
Given the expression for the force of Eq. (S10) (Eq. (1) of the main text), we evaluate here the mechanical frequency shift for a mechanical resonator whose electrical properties can be described by Eq. (S1).
For sake of simplicity, we focus here on the analysis of a membrane (vanishing flexural rigidity) in the presence of uniform tension. The only difference with the general case (finite flexural rigidity and nonuniform tension) is the specific value of the mechanical frequency in absence of external applied voltage ω 0,n , which is not central to our argument.
Firstly, we notice that the distance between the plates of the capacitor -and, consequently, the capacitance C g -is modified by the application of the external electrostatic potential V g (see Fig. S1). The equilibrium position of the movable capacitance ξ e = ξ e (V g , B) can be obtained from the solution of the general elastic equation for the structure (see e.g. 5 for the case of a graphene-only membrane).
The shifts of the mechanical resonant frequency ω 0,n for V g ̸ = 0 can be determined considering the membrane position fluctuations around ξ e . For a membrane, these fluctuations obey the following equation where we have defined δf = [f (ξ e + ζ) − f (ξ e )] and T = T (ξ e ). The functions ζ(r, t) and δf (r, t) in Eq. (S11) can be expanded as where A(r) are the normal modes associated with the boundary value problem considered.
Furthermore, approximating we obtain leading to the following expression for the frequency of a given mode n with ω 0,n = T λ 2 n /ρ, where λ n depends on the geometry considered. For the case of a disk of radius R, we have that λ n → λ i,j = α i,j /R, where α i,j are the roots of the Bessel function J i,j (r). For the case of metallic leads (infinite DOS), we have that ∂F/∂ζ) = 1/2C ′′ g V 2 g allowing us to obtain from Eq. (S13) the usual capacitive softening voltage dependence of the mechanical resonant frequency.
In the case of finite DOS, from Eq. (S10), we can write which with the help of Eq. (S6), gives Denoting by ∆ω n the difference between the resonant frequency at finite B with respect to the frequency at B = 0, we can write where ∆ω 0,n = ω 0,n (B) − ω 0,n (B = 0). When writing ∂F/∂ζ| B , we have considered both the the explicit B−dependence of the derivative at ξ e = ξ e (0), and the dependence of ξ e on the magnetic field. The first term in Eq. (S16) represents the change of the resonant frequency with the external magnetic field associated with tensioning effects, whereas the second term is related to the magnetic field change of the capacitive softening term.
Noting that, for B = 0, the resonant frequency ω 0,n depends on the externally applied voltage only, we can approximate (S17) Integrating equation (S6), in the limit V g ≫ µ/e and C q ≫ C g , we have The two terms in Eq. (S18) take into account the explicit dependence of µ on the external magnetic field and its dependence on ξ e , respectively.
Equations (S17, S18) lead to Analogously, we have that leading to which with κ n = ρω 2 n and ω n = 2πf n leads to Eq. (2) of the main text. We note also that, in the limit C ′′ g = 0, Eq. (S21) corresponds to the expression given in 6 for the mechanical frequency shift.

Quantum capacitance
We derive here the expression for the quantum capacitance C q . To this end, we express the density of states as a sum of Lorentzians centered at with the Landau level degeneracy factor N = 4. Our analysis is based on the possibility of turning an infinite sum into an integral over the complex plane. If f (w) is a meromorphic function, the following condition is fulfilled Since, in our case, we have we operate a change of variables w → z 2 , which allows us to rewrite Eq. (S23) as . The red line represents the contour along which the integral on the lhs is performed. We have denoted the poles of 2πz cot(πz) (whose residue is f (n)) in blue, whereas the poles of f (z 2 ) are marked in red.
in Ref. 7 Conversely, the second term, the sum over the poles of 2πz cot(πz)f (z), can be evaluated explicitly, leading to the following expression for the density of states where Λ D is the contribution coming from the integral on the contour C n .
In the ω ≫ γ, ω D limit the expression for D(ω) given in Eq. (S26), coincides with the one given in Ref., 7 in the same limit. From Eq. (S26) we can easily derive the expression for

Quantum capacitance and de Haas -van Alphen effect
With an analogous calculation to the one leading to the expression for C q , it is possible to derive an expression for the charge density n and to the oscillatory component of the magnetization The relations given by Eqs. (S27,S28aS28b, complemented by the relation allow us to determine (numerically, for finite values of C g ) the dependence of µ and n on µ = eV g . For C g → 0 and C g → ∞, as mentioned in the main text, it is possible to consider the charge n or the chemical potential µ as control parameters for the graphene disk (see also 7 ). From Eqs. (S28aS28b, S29) is straightforward to demonstrate a relation between the oscillating component of the quantum capacitance C q,osc = e 2 ∂n osc /∂µ and the oscillations of the magnetic susceptibility χ m,osc = ∂Mosc ∂B . From Eqs. (S28a-S29), we have that which leads to the relation Here for µ ≫ γ, we obtain (S34) We note here that Eq. (S32) can be obtained from a general thermodynamic argument.
Considering the two pairs of intensive/extensive variables (n, µ), (M, B), we can define four possible thermodynamic potentials. We focus here on the choice G = G(n, B), which allows us to define From the chain rule of the derivative, considering that µ and M can be intepreted as independent variables -this is in fact the choice we have operated above when considering the thermodynamics potential Ω = Ω(µ, M )-we have allowing us to write which, considering the Maxwell relation ∂M/∂n| µ = − ∂µ/∂B| n , leads to Eq. (S38) establishes a general relation between quantum capacitance and magnetic susceptibility, which, for graphene, in the limit µ ≫ γ, corresponds to the expression given in Eq. (S34).
Neglecting the small oscillations of the chemical potential, it is therefore clear that the dips in the dependence of the mechanical resonant frequency ω n as a function of B, which were previously been interpreted in terms of quantum capacitance oscillations, can equivalently be interpreted in terms of oscillations of the magnetic susceptibility χ m (de Haas -van Alphen effect). From Eq. (S32) we can write Substituting Eq. (S39) into Eq. (S21) we obtain 2ρω 0,n χ m,osc /Γ C q0 (C q0 + χ m,osc /Γ) (S40) which, upon defining 2ρω 0,n η = C q0 Γ, leads to Eq. (2) of the main text, where C q,0 = e 2 ∂n 0 /∂µ = N e 2 µ/(πv 2 F ℏ 2 ). Close to the frequency dips, Λ 1 ≈ 0 and, given the large elastic constant of gold, From the value of Λ 2 it is possible to establish the optimal gate voltage for the observation of the frequency dips, which is given by

Sample fabrication and measurement setting
Our Au resonators with suspended graphene Corbino disks were fabricated using a method adapted from Ref. 8  The fabricated devices were characterized using standard conductance measurement techniques and resonance measurements at 10 mK. The devices were mounted slightly off center of the 9 T magnet on a Bluefors LD400 dilution refrigerator. At the sample location, dB/dz = 60 T/m and d 2 B/dz 2 = 1100 T/m 2 with a maximum field of 6.8 T. This second derivative has such a tiny effect on the Au mechanical frequencies so that it can be neglected in our force analysis.
Prior to the actual measurements, however, current annealing 10 was performed by applying a bias voltage V b ≈ 2 V across the Corbino ring, consequently evaporating residues from fabrication off from the graphene flake. The device quality was assessed by measuring the Landau fan diagram, such diagram is presented in Fig. 2a for the investigated sample B2.
Note the fractional QH state ν = 1/3 is visible from B ≈ 3 T upwards along with the usual set of integer quantum Hall states highlighting the good quality of the measured samples.
At higher fields more fractional states appeared, see Refs. 9,11 .

Identification of quantum Hall states
Detection of the QH states in the graphene Corbino was performed both using low-frequency AC conductance and the mechanical response of the combined gold-graphene modes. The sensitive Au resonance detection of QH states via graphene's mechanical response is facilitated by the variation of the derivative dG/dV g that specifies the magnitude of the mixing current I mix in graphene (see Eq. (S42)). Consequently, I mix pinpoints regions with dG/dV g = 0, across which the mixing current changes its sign. In the experiment, the sign change of I mix is seen as a flip of the phase by π in the down-mixed signal. Fig. 3 displays the measured conductance G(V g ) and the phase of the mixing current. An exact match between dG/dV g = 0 locations and the phase flips is observed.

Mechanical resonances
Mechanical resonances were detected using the FM mixing technique. 12 In this technique where C tot = (1/C g + 1/C q ) −1 is the total capacitance, and C ′ tot = dCtot dz . C g and C q are the gate capacitance and the quantum capacitance per unit area, respectively. Phase shifts may occur between the drive and the response due interference phenomena in the flexural waves traveling along the Corbino disk, driven from the outer edge. In the case of phase shifts, the mechanical response function ∂Re(z) ∂f will obtain a corresponding reference phase, which results in a combination of dispersive and absorptive parts of the mechanical response. The downmixed signal from the other electrode of the corbino disk was led back to the lock-in amplifier through a Stanford SR570 current amplifier with gain 10 6 V/A. In our frequency sweeps of the sample B1.5, we observed 12 mechanical modes below 27 MHz. Using COMSOL simulations, candidate mode shapes for these modes could be identified. We utilized simulated gate voltage dependencies for each mode to determine the mode shape corresponding to the 26.5 MHz resonance with which the dHvA effect was observed. A mode shape, where the most significant role is played by the cantilever, displays a weak frequency increase with respect to the gate voltage in the simulations corresponding to the observed trend in the measurements.
We emphasize that our detection scheme for magnetization effects in a 2D material relies on finding well-defined resonances of the gold structure and is, therefore, suitable for a very wide variety 2D systems. Specifically, the 2D-material portion of the structure is not required In addition to graphene and Au modes, surface waves around 20 MHz were excited in the LOR-layer by the microwave drive. 13 Even though these modes could be excited at very small power, they were not useful for detection purposes owing to the small Q = 100 − 200.

Fitting the theory predictions
The theoretical predictions of the frequency shift due to the dHvA effect given in Eq. 2 are fitted to the experimental data as shown in Figs. 4a and 4b. To obtain the spring constant of the modes, we assume that the effective mass of the modes is determined by the gold resonator, and graphene's contribution is practically negligible. We approximate that, for the B2 device ∼ 80 nm thick lower gold beam, the 2D density is ρ ≈ 1.5 × 10 −3 kg/m 2 , and ρ ≈ 2.1 × 10 −3 kg/m 2 for the ∼ 110 nm thick cantilever of B1.5.
For our setup, the last term of the frequency shift in Eq. 2 is negligible and, therefore, the size of the frequency shift is scaled by the factor ∂f 0,n /∂V . We fit ∂f 0,n /∂V to the experimental data to obtain proper magnitude of ∆f together with the Landau level width γ that affects the magnitude of ∆f as well as the width of the frequency dips. For the device B2 (B1.5) we have 16 kHz/V (35 kHz/V). The larger value for the B1.5 device mode is expected due to it being a mode of the cantilever whose other end is attached to the graphene. This boundary condition makes the B1.5 mode more sensitive to tensioning effects than the resonance of the lower gold plate of the B2 device.
The scattering time τ S = ℏ √ πnµ q /ev F shown in Fig. 3c was calculated from the quantum mobility µ q , which in turn was obtained by extracting the minimum field of Shubnikov-de Haas oscillations B 0 (V g ) and using the relation µ q B 0 = 1. Additionally, the error bars of τ q in the same figure show a 15% deviation from the values of γ = ℏ/(2τ q ) used to fit the theory curves in Fig. 3a. Values of γ within this tolerance reproduce a good agreement of ∆f between the theory and the experiment.